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Sungkyung Kang

Publications and source records attributed to Sungkyung Kang.

At least 19 recordsLinked to original sources
arXiv2026

$\widetilde{H}$-cobordisms, infinite cyclic covers, and real Seiberg--Witten theory

We study $\widetilde{H}$-cobordisms of distinguished homology handles, introduced by Kawauchi in 1976 using infinite cyclic covers. Despite the extensive development of gauge-theoretic and Floer-theoretic invariants since Kawauchi's work, none were previously known to distinguish smooth and topological $\widetilde{H}$-cobordism. In this paper, we construct asymptotic invariants of distinguished homology handles by applying real Seiberg--Witten theory to finite cyclic covers. Using these invariants, we show that the kernel of the natural map from the smooth $\widetilde{H}$-cobordism group to its topological counterpart contains a subgroup isomorphic to $\mathbb{Z}$. We also define spin versions of these groups and show that the kernel of the corresponding natural map contains a subgroup isomorphic to $\mathbb{Z}^\infty$.

math.GT↗
arXiv2026

Invariant splitting principles for the Lipshitz--Ozsváth--Thurston correspondence

We prove that the Lipshitz-Ozsváth-Thurston correspondence between extended type D structures of knot complements and $\mathbb{F}[U, V]/(UV)$ knot Floer complexes can be arranged so that $ι_K$-invariant splittings of knot Floer chain complexes correspond to $ι_{S^3 \setminus K}$-invariant splittings of bordered Floer homology of knot complements. For patterns satisfying the satellite extension property, which include cabling patterns, this provides a novel way to compute the involutive knot Floer homology of satellites from that of their companions. As a topological application, we show that our results can be applied to construct infinitely many examples of exotic pairs of contractible 4-manifolds which remain exotic after one stabilization. Along the way, we also establish first order naturality of bordered Floer homology.

math.GT↗
arXiv2026

Satellites and telescopes: a concordance formula for bordered Floer homology

Given a knot $K$ in $S^3$, its knot Floer completely determines the bordered Floer homology of its complement by a work of Lipshitz--Ozsváth--Thurston. Furthermore, the determination is combinatorial: given a model for $CFK(S^3, K)$ there is a method for producing an explicit model for $\widehat{CFD}(S^3 \smallsetminus ν(K))$. In this paper, we show that a similar formula holds between certain classes of chain endomorphisms of knot Floer chain complex and type D endomorphisms of bordered Floer homology, up to a 1-dimensional ambiguity in the type D side; both the formula and the ambiguity can be computed combinatorially. It follows that, for any concordance from a knot to itself and any satellite pattern, we can combinatorially compute the knot Floer cobordism map of the satellite concordance (up to conjugation) from the knot Floer cobordism map of the given concordance and the bordered Floer homology of the pattern complement.

math.GT↗
arXiv2026

Non-kinetic homotopy coherent actions on four-manifolds

We give the first example of a non-kinetic smooth homotopy coherent action of order two on a closed simply connected smooth four-manifold. This action is obtained by restricting a nontrivial smooth homotopy coherent action of the discrete circle group on a stabilized $K3$ surface. We also construct relatively non-kinetic smooth homotopy coherent extensions of boundary involutions over compact smooth $4$-manifolds. In addition, we exhibit boundary involutions that admit locally linear topological extensions but no smooth extensions over the same stabilized fillings. Nevertheless, we prove a Wall-type theorem showing that every free involution on a disjoint union of integral homology spheres extends smoothly over any simply connected smooth filling after sufficiently many stabilizations by $S^2\times S^2$.

math.GT↗
arXiv2025

Smooth concordance of cables of the figure-eight knot

We prove that every nontrivial cable of the figure-eight knot has infinite order in the smooth knot concordance group. Our main contribution is a uniform proof that applies to all $(2n,1)$-cables of the figure-eight knot. To this end, we introduce a family of concordance invariants $κ_R^{(k)}$, defined via $2^k$-fold branched covers and real Seiberg--Witten Floer $K$-theory. These invariants generalize the real $K$-theoretic Frøyshov invariant developed by Konno, Miyazawa, and Taniguchi.

math.GT↗
arXiv2025

Cables of the figure-eight knot via real Frøyshov invariants

We prove that the $(2n,1)$-cable of the figure-eight knot is not smoothly slice when $n$ is odd, by using the real Seiberg-Witten Frøyshov invariant of Konno-Miyazawa-Taniguchi. For the computation, we develop an $O(2)$-equivariant version of the lattice homotopy type, originally introduced by Dai-Sasahira-Stoffregen. This enables us to compute the real Seiberg-Witten Floer homotopy type for a certain class of knots. Additionally, we present some computations of Miyazawa's real framed Seiberg-Witten invariant for 2-knots.

math.GT↗
arXiv2024

Exotic Dehn twists and homotopy coherent group actions

We consider the question of extending a smooth homotopy coherent finite cyclic group action on the boundary of a smooth 4-manifold to its interior. As a result, we prove that Dehn twists along any Seifert homology sphere, except the 3-sphere, on their simply connected positive-definite fillings are infinite order exotic.

math.GT↗
arXiv2024

One stabilization is not enough for contractible 4-manifolds

We construct an example of a cork that remains exotic after taking a connected sum with $S^2 \times S^2$. Combined with a work of Akbulut-Ruberman, this implies the existence of an exotic pair of contractible 4-manifolds which remains absolutely exotic after taking a connected sum with $S^2 \times S^2$.

math.GT↗
arXiv2024

On the nonorientable four-ball genus of torus knots

The nonorientable four-ball genus of a knot $K$ in $S^3$ is the minimal first Betti number of nonorientable surfaces in $B^4$ bounded by $K$. By amalgamating ideas from involutive knot Floer homology and unoriented knot Floer homology, we give a new lower bound on the smooth nonorientable four-ball genus $γ_4$ of any knot. This bound is sharp for several families of torus knots, including $T_{4n,(2n\pm 1)^2}$ for even $n\ge 2$, a family Longo showed were counterexamples to Batson's conjecture. We also prove that, whenever $p$ is an even positive integer and $\frac{p}{2}$ is not a perfect square, the torus knot $T_{p,q}$ does not bound a locally flat Möbius band for almost all integers $q$ relatively prime to $p$.

math.GT↗
arXiv2023

Central extensions of higher groups: Green-Schwarz mechanism and 2-connections

We study the smooth $2$-group structure arising in the presence of quantum field theory with one-form symmetry. We acquire $2$-group structures obtained by a central extension of the zero-form symmetry by the one-form symmetry. We determine that the existence of a $2$-group structure is guaranteed by Chern--Simons levels. We further verify how we will be able to provide a fix to the current $2$-group problems by using the bibundle model. We outline the principal $2$-connection theory with respect to such $2$-group and compare it with the ansatz obtained from the Green--Schwarz mechanism. We further propose the existence of smooth $\infty$-group symmetries in quantum field theory.

hep-th↗
arXiv2023

Doubled Disks and Satellite Surfaces

Conjecturally, a knot is slice if and only if its positive Whitehead double is slice. We consider an analogue of this conjecture for slice disks in the four-ball: two slice disks of a knot are smoothly isotopic if and only if their positive Whitehead doubles are smoothly isotopic. We provide evidence for this conjecture, using a range of techniques. More generally, we consider when isotopy obstructions persist under satellite operations. In particular, we show that obstructions coming from knot Floer homology, Seiberg-Witten theory, and Khovanov homology often behave well under satellite operations. We apply these strategies to give a systematic method for constructing vast numbers of exotic disks in the four-ball, including the first infinite family of pairwise exotic slice disks. These same techniques are then upgraded to produce exotic disks that remain exotic after any prescribed number of internal stabilizations. Finally, we show that the branched double covers of certain stably-exotic disks become diffeomorphic after a single stabilization with $S^2 \times S^2$, hence stabilizing them yields exotic surfaces that have diffeomorphic branched covers.

math.GT↗
arXiv2023

Topologically and rationally slice knots

A knot in $S^3$ is topologically slice if it bounds a locally flat disk in $B^4$. A knot in $S^3$ is rationally slice if it bounds a smooth disk in a rational homology ball. We prove that the smooth concordance group of topologically and rationally slice knots admits a $\mathbb{Z}^\infty$ subgroup. All previously known examples of knots that are both topologically and rationally slice were of order two. As a direct consequence, it follows that there are infinitely many topologically slice knots that are strongly rationally slice but not slice.

math.GT↗
arXiv2023

One stabilization is not enough for closed knotted surfaces

In this brief note, we show that there exist smooth 4-manifolds (with nonempty boundary) containing pairs of exotically knotted 2-spheres that remain exotic after one (either external or internal) stabilization. It follows that the ``one is enough'' theorem of Auckly-Kim-Melvin-Ruberman-Schwartz does not hold for closed surfaces whose homology classes are characteristic.

math.GT↗
arXiv2022

Torsion in the knot concordance group and cabling

We define a nontrivial mod 2 valued additive concordance invariant defined on the torsion subgroup of the knot concordance group using involutive knot Floer package. For knots not contained in its kernel, we prove that their iterated $(\text{odd},1)$-cables have infinite order in the concordance group and, among them, infinitely many are linearly independent. Furthermore, by taking $(2,1)$-cables of the aforementioned knots, we present an infinite family of knots which are strongly rationally slice but not slice.

math.GT↗
arXiv2022

Involutive knot Floer homology and bordered modules

We prove that, up to local equivalences, a suitable truncation of the involutive knot Floer homology of a knot in $S^3$ and the involutive bordered Heegaard Floer theory of its complement determine each other. In particular, given two knots $K_1$ and $K_2$, we prove that the $\mathbb{F}_2[U,V]/(UV)$-coefficient involutive knot Floer homology of $K_1 \sharp -K_2$ is $ι_K$-locally trivial if $\widehat{CFD}(S^3 \backslash K_1)$ and $\widehat{CFD}(S^2 \backslash K_2)$ satisfy a certain condition which can be seen as the bordered counterpart of $ι_K$-local equivalence. We further establish an explicit algebraic formula that computes the hat-flavored truncation of the involutive knot Floer homology of a knot from the involutive bordered Floer homology of its complement. It follows that there exists an algebraic satellite operator defined on the local equivalence group of knot Floer chain complexes, which can be computed explicitly up to a suitable truncation.

math.GT↗
arXiv2021

Linear independence of rationally slice knots

A knot in $S^3$ is rationally slice if it bounds a disk in a rational homology ball. We give an infinite family of rationally slice knots that are linearly independent in the knot concordance group. In particular, our examples are all infinite order. All previously known examples of rationally slice knots were order two.

math.GT↗